Method of and apparatus for independently determining the resistivity and/or dielectric constant of an earth formation

ABSTRACT

Techniques are provided to transform attenuation and phase measurements taken in conjunction with a drilling operation into independent electrical parameters such as electrical resistivity and dielectric values. The electrical parameters are correlated with background values such that resulting estimates of the electrical parameters are independent of each other. It is shown an attenuation measurement is sensitive to the resistivity in essentially the same volume of an earth formation as the corresponding phase measurement is sensitive to the dielectric constant. Further, the attenuation measurement is shown to be sensitive to the dielectric constant in essentially the same volume that the corresponding phase measurement is sensitive to the resistivity. Techniques are employed to define systems of simultaneous equations that produce more accurate measurements of the resistivity and/or the dielectric constant within the earth formation than are available from currently practiced techniques.

RELATED APPLICATIONS

This continuation application claims priority to U.S. patent applicationSer. No. 09/608,205, filed on Jun. 30, 2000, now U.S. Pat. No.6,366,858.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to a method of surveying earthformations in a borehole and, more specifically, to a method of andapparatus for independently determining the electrical resistivityand/or dielectric constant of earth formations during Measurement-WhileDrilling/Logging-While-Drilling and Wireline Logging operations.

2. Description of the Related Art

Typical petroleum drilling operations employ a number of techniques togather information about earth formations during and in conjunction withdrilling operations such as Wireline Logging, Measurement-While-Drilling(MWD) and Logging-While-Drilling (LWD) operations. Physical values suchas the electrical conductivity and the dielectric constant of an earthformation can indicate either the presence or absence of oil-bearingstructures near a drill hole, or “borehole.” A wealth of otherinformation that is useful for oil well drilling and production isfrequently derived from such measurements. Originally, a drill pipe anda drill bit were pulled from the borehole and then instruments wereinserted into the hole in order to collect information about down holeconditions. This technique, or “wireline logging,” can be expensive interms of both money and time. In addition, wireline data may be of poorquality and difficult to interpret due to deterioration of the regionnear the borehole after drilling. These factors lead to the developmentof Logging-While-Drilling (LWD). LWD operations involve collecting thesame type of information as wireline logging without the need to pullthe drilling apparatus from the borehole. Since the data are taken whiledrilling, the measurements are often more representative of virginformation conditions because the near-borehole region often deterioratesover time after the well is drilled. For example, the drilling fluidoften penetrates or invades the rock over time, making it more difficultto determine whether the fluids observed within the rock are naturallyoccurring or drilling induced. Data acquired while drilling are oftenused to aid the drilling process. For example, MWD/LWD data can help adriller navigate the well so that the borehole is ideally positionedwithin an oil bearing structure. The distinction between LWD and MWD isnot always obvious, but MWD usually refers to measurements taken for thepurpose of drilling the well (such as navigation) whereas LWD isprincipally for the purpose of estimating the fluid production from theearth formation. These terms will hereafter be used synonymously andreferred to collectively as “MWD/LWD.”

In wireline logging, wireline induction measurements are commonly usedto gather information used to calculate the electrical conductivity, orits inverse resistivity. See for example U.S. Pat. No. 5,157,605. Adielectric wireline tool is used to determine the dielectric constantand/or resistivity of an earth formation. This is typically done usingmeasurements which are sensitive to the volume near the borehole wall.See for example U.S. Pat. No. 3,944,910. In MWD/LWD, a MWD/LWDresistivity tool is typically employed. Such devices are often called“propagation resistivity” or “wave resistivity” tools, and they operateat frequencies high enough that the measurement is sensitive to thedielectric constant under conditions of either high resistivity or alarge dielectric constant. See for example U.S. Pat. Nos. 4,899,112and4,968,940. In MWD applications, resistivity measurements may be used forthe purpose of evaluating the position of the borehole with respect toboundaries of the reservoir such as with respect to a nearby shale bed.The same resistivity tools used for LWD may also used for MWD; but, inLWD, other formation evaluation measurements including density andporosity are typically employed.

For purposes of this disclosure, the terms “resistivity” and“conductivity” will be used interchangeably with the understanding thatthey are inverses of each other and the measurement of either can beconverted into the other by means of simple mathematical calculations.The terms “depth,” “point(s) along the borehole,” and “distance alongthe borehole axis” will also be used interchangeably. Since the boreholeaxis may be tilted with respect to the vertical, it is sometimesnecessary to distinguish between the vertical depth and distance alongthe borehole axis. Should the vertical depth be referred to, it will beexplicitly referred to as the “vertical depth.”

Typically, the electrical conductivity of an earth formation is notmeasured directly. It is instead inferred from other measurements eithertaken during (MWD/LWD) or after (Wireline Logging) the drillingoperation. In typical embodiments of MWD/LWD resistivity devices, thedirect measurements are the magnitude and the phase shift of atransmitted electrical signal traveling past a receiver array. See forexample U.S. Pat. Nos. 4,899,112, 4,968,940, or 5,811,973. In commonlypracticed embodiments, the transmitter emits electrical signals offrequencies typically between four hundred thousand and two millioncycles per second (0.4-2.0 MHz). Two induction coils spaced along theaxis of the drill collar having magnetic moments substantially parallelto the axis of the drill collar typically comprise the receiver array.The transmitter is typically an induction coil spaced along the axis ofa drill collar from the receiver with its magnetic moment substantiallyparallel to the axis of the drill collar. A frequently used mode ofoperation is to energize the transmitter for a long enough time toresult in the signal being essentially a continuous wave (only afraction of a second is needed at typical frequencies of operation). Themagnitude and phase of the signal at one receiving coil is recordedrelative to its value at the other receiving coil. The magnitude isoften referred to as the attenuation, and the phase is often called thephase shift. Thus, the magnitude, or attenuation, and the phase shift,or phase, are typically derived from the ratio of the voltage at onereceiver antenna relative to the voltage at another receiver antenna.

Commercially deployed MWD/LWD resistivity measurement systems usemultiple transmitters; consequently, attenuation and phase-basedresistivity values can be derived independently using each transmitteror from averages of signals from two or more transmitters. See forexample U.S. Pat. No. 5,594,343.

As demonstrated in U.S. Pat. Nos. 4,968,940 and 4,899,112, a very commonmethod practiced by those skilled in the art of MWD/LWD for determiningthe resistivity from the measured data is to transform the dielectricconstant into a variable that depends on the resistivity and then toindependently convert the phase shift and attenuation measurements totwo separate resistivity values. A key assumption implicitly used inthis practice is that each measurement senses the resistivity within thesame volume that it senses the dielectric constant. This implicitassumption is shown herein by the Applicant to be false. This currentlypracticed method may provide significantly incorrect resistivity values,even in virtually homogeneous earth formations; and the errors may beeven more severe in inhomogeneous formations.

A MWD/LWD tool typically transmits a 2 MHz signal (although frequenciesas low as 0.4 MHz are sometimes used). This frequency range is highenough to create difficulties in transforming the raw attenuation andphase measurements into accurate estimates of the resistivity and/or thedielectric constant. For example, the directly measured values are notlinearly dependent on either the resistivity or the dielectric constant(this nonlinearity, known to those skilled in the art as “skin-effect,”also limits the penetration of the fields into the earth formation). Inaddition, it is useful to separate the effects of the dielectricconstant and the resistivity on the attenuation and phase measurementsgiven that both the resistivity and the dielectric constant typicallyvary spatially within the earth formation. If the effects of both ofthese variables on the measurements are not separated, the estimate ofthe resistivity can be corrupted by the dielectric constant, and theestimate of the dielectric constant can be corrupted by the resistivity.Essentially, the utility of separating the effects is to obtainestimates of one parameter that don't depend on (are independent of) theother parameter. A commonly used current practice relies on assuming acorrelative relationship between the resistivity and dielectric constant(i.e., to transform the dielectric constant into a variable that dependson the resistivity) and then calculating resistivity valuesindependently from the attenuation and phase shift measurements that areconsistent the correlative relationship. Differences between theresistivity values derived from corresponding phase and attenuationmeasurements are then ascribed to spatial variations (inhomogeneities)in the resistivity over the sensitive volume of the phase shift andattenuation measurements. See for example U.S. Pat. Nos. 4,899,112 and4,968,940. An implicit and instrumental assumption in this method isthat the attenuation measurement senses both the resistivity anddielectric constant within the same volume, and that the phase shiftmeasurement senses both variables within the same volume (but not thesame volume as the attenuation measurement). See for example U.S. Pat.Nos. 4,899,112 and, 4,968,940. These assumptions facilitate theindependent determination of a resistivity value from a phasemeasurement and another resistivity value from an attenuationmeasurement. However, as is shown later, the implicit assumptionmentioned above is not true; so, the results determined using suchalgorithms are questionable. Methods are herein disclosed to determinetwo resistivity values from a phase and an attenuation measurement donot use the false assumptions of the above mentioned prior art.

Another method for determining the resistivity and/or dielectricconstant is to assume a model for the measurement apparatus in, forexample, a homogeneous medium (no spatial variation in either theresistivity or dielectric constant) and then to determine values for theresistivity and dielectric constant that cause the model to agree withthe measured phase shift and attenuation data. The resistivity anddielectric constant determined by the model are then correlated to theactual parameters of the earth formation. This method is thought to bevalid only in a homogeneous medium because of the implicit assumptionmentioned in the above paragraph. A recent publication by P. T. Wu, J.R. Lovell, B. Clark, S. D. Bonner, and J. R. Tabanou entitled“Dielectric-Independent 2-MHz Propagation Resistivities” (SPE 56448,1999) (hereafter referred to as “Wu”) demonstrates that such assumptionsare used by those skilled in the art. For example, Wu states: “Onefundamental assumption in the computation of Rex is an uninvadedhomogeneous formation. This is because the phase shift and attenuationinvestigate slightly different volumes.” It is shown herein by Applicantthat abandoning the false assumptions applied in this practice resultsin estimates of one parameter (i.e., the resistivity or dielectricconstant) that have no net sensitivity to the other parameter. Thisdesirable and previously unknown property of the results is very usefulbecause earth formations are commonly inhomogeneous.

Wireline dielectric measurement tools commonly use electrical signalshaving frequencies in the range 20 MHz-1.1 GHz. In this range, theskin-effect is even more severe, and it is even more useful to separatethe effects of the dielectric constant and resistivity. Those skilled inthe art of dielectric measurements have also falsely assumed that ameasurement (either attenuation or phase) senses both the resistivityand dielectric constant within the same volume. The design of themeasurement equipment and interpretation of the data both reflect this.See for example U.S. Pat. Nos. 4,185,238 and 4,209,747.

Wireline induction measurements are typically not attenuation and phase,but instead the real (R) and imaginary (X) parts of the voltage across areceiver antenna which consists of several induction coils in electricalseries. For the purpose of this disclosure, the R-signal for a wirelineinduction measurement corresponds to the phase measurement of a MWD/LWDresistivity or wireline dielectric tool, and the X-signal for a wirelineinduction measurement corresponds to the attenuation measurement of aMWD/LWD resistivity or wireline dielectric device. Wireline inductiontools typically operate using electrical signals at frequencies from8-200 kHz (most commonly at approximately 20 kHz). This frequency rangeis too low for significant dielectric sensitivity in normallyencountered cases; however, the skin-effect can corrupt the wirelineinduction measurements. As mentioned above, the skin-effect shows up asa non-linearity in the measurement as a function of the formationconductivity, and also as a dependence of the measurement sensitivityvalues on the formation conductivity. Estimates of the formationconductivity from wireline induction devices are often derived from dataprocessing algorithms which assume the tool response function is thesame at all depths within the processing window. The techniques of thisdisclosure can be applied to wireline induction measurements for thepurpose of deriving resistivity values without assuming the toolresponse function is the same at all depths within the processing windowas is done in U.S. Pat. No. 5,157,605. In order to make such anassumption, a background conductivity, σ, that applies for the datawithin the processing window is commonly used. Practicing a disclosedembodiment reduces the dependence of the results on the accuracy of theestimates for the background parameters because the backgroundparameters are not required to be the same at all depths within theprocessing window. In addition, practicing appropriate embodiments ofApplicant's techniques discussed herein reduces the need to performsteps to correct wireline induction data for the skin effect.

SUMMARY OF THE INVENTION

Techniques are provided to transform attenuation and phase measurementstaken in conjunction with a drilling operation into quantities suitablefor producing more accurate electrical conductivity and/or dielectricconstant values. The electrical conductivity and dielectric constantvalues are interpreted to provide information such as the presence orabsence of hydrocarbons within an earth formation penetrated by thedrilling operation. The techniques can be applied to Wireline Logging,Logging-While-Drilling (LWD) and Measurement-While-Drilling (MWD)operations.

As explained above, current data processing practices in the field ofMWD/LWD and wireline dielectric logging are based upon the assumptionthat an attenuation measurement is sensitive to the resistivity value ofan earth formation in the same volume as the attenuation is sensitive tothe dielectric constant. Current data processing practices are alsobased upon the assumption that the phase measurement is sensitive to theresistivity in the same volume of the earth formation as it is sensitiveto the dielectric constant, but that this volume of the phasemeasurement is different from that of the attenuation measurement. Theseassumptions, referred to herein as the “old assumptions,” are shown tobe false. In fact, the attenuation senses the resistivity and thedielectric constant in different volumes; and the phase shift senses theresistivity and the dielectric constant in different volumes. However,the attenuation measurement is shown to be sensitive to the resistivityin essentially the same volume as the phase measurement is sensitive tothe dielectric constant. Further, the attenuation measurement is shownto be sensitive to the dielectric constant in essentially the samevolume that the phase measurement is sensitive to the resistivity.

By employing these new-found relationships among the attenuation, phase,resistivity and dielectric constant, systems of simultaneous equationsare provided that produce more accurate measurements of the resistivityand/or the dielectric constant within an earth formation thanmeasurements produced using the old assumptions. In some embodiments,the equations are manipulated in a manner that provides a resistivitycomponent that is relatively insensitive to the dielectric constant andprovides a dielectric constant component that is relatively insensitiveto the resistivity value. Thus, more accurate and/or robust calculationsof both the resistivity and the dielectric constant are produced.

The disclosed techniques are also applied to more complicated scenarioswherein multiple transmitters (possibly driven at multiple frequencies),multiple receivers, data acquired at multiple depths, or combinations ofthe above are considered simultaneously. Solving a prescribed system ofequations using this disclosed embodiment results in estimates of anaverage conductivity value and an average dielectric constant valuewithin a volume of the earth formation corresponding to integratedaverages of each parameter over said volume. In general, the resistivityand dielectric constant are expanded using basis functions tocharacterize the spatial dependence of these variables. A system ofequations which can be solved for the coefficients of this expansion isgiven. Once the coefficients are determined, the spatial dependence ofboth the resistivity and dielectric constant are known.

One disclosed embodiment employs a transformation to convert thedielectric constant into a variable that depends on the resistivitythereby eliminating the dielectric constant as a variable. Tworesistivity estimates from a phase shift and an attenuation measurementare then calculated. These estimates are not determined independently asis done in the prior art because the equations solved to obtain theestimates are coupled. The manner in which these equations are coupledis consistent with the actual sensitivities of each measurement (i.e.,the phase shift and attenuation) with respect to changes in eachvariable (i.e., the resistivity and dielectric constant). Unlikeprevious MWD/LWD processing techniques, some disclosed embodimentsaccount for dielectric effects, provide for inhomogeneities, and treateach signal as a complex-valued function of the conductivity anddielectric constant, assuring that estimates of each variable are notcorrupted by effects of the other. Treating both the measurements(attenuation and phase) and the variables (conductivity and dielectricconstant) mathematically as complex-valued functions is a useful featureof the disclosed embodiments. Good results from the disclosedembodiments are produced by using the new found relationship regardingthe volume of investigation of each measurement with respect to theconductivity and the dielectric constant. In contrast, the oldassumptions imply that these results are impracticable. This is readilyevident from discussions in U.S. Pat. Nos. 4,185,238; 4,209,747;4,899,112; and 4,968,940.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description of some preferred embodiments isconsidered in conjunction with the following drawings, in which:

FIG. 1 is a plot of multiple laboratory measurements on rock samplesrepresenting the relationship between the conductivity and thedielectric constant in a variety of geological media;

FIG. 2 illustrates the derivation of a sensitivity function in relationto an exemplary one-transmitter, one-receiver MWD/LWD resistivity tool;

FIG. 3 illustrates an exemplary one-transmitter, two-receiver MWD/LWDtool commonly referred to as an uncompensated measurement device;

FIG. 4 illustrates an exemplary two-transmitter, two-receiver MWD/LWDtool, commonly referred to as a compensated measurement device;

FIGS. 5a, 5 b, 5 c and 5 d are exemplary sensitivity function plots forDeep and Medium attenuation and phase shift measurements;

FIGS. 6a, 6 b, 6 c and 6 d are plots of the sensitivity functions forthe Deep and Medium measurements of FIGS. 5a, 5 b, 5 c and 5 drespectively transformed according to the techniques of a disclosedembodiment;

FIG. 7 is a portion of a table of background medium values and integralvalues employed in a disclosed embodiment;

FIG. 8 is a plot of attenuation and phase as a function of resistivityand dielectric constant; and

FIG. 9 is a flowchart of a process, that implements the techniques of adisclosed embodiment.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Some of the disclosed embodiments are relevant to both wirelineinduction and dielectric applications, as well asMeasurement-While-Drilling and Logging-While-Drilling (MWD/LWD)applications. Turning now to the figures, FIG. 1 is a plot ofmeasurements of the conductivity and dielectric constant determined bylaboratory measurements on a variety of rock samples from differentgeological environments. The points 121 through 129 represent measuredvalues of conductivity and dielectric constant (electrical parameters)for carbonate and sandstone earth formations. For instance, the point126 represents a sample with a conductivity value of 0.01 (10⁻²) siemensper meter (S/m) and a relative dielectric constant of approximately 22.It should be noted that both the conductivity scale and the dielectricscale are logarithmic scales; so, the data would appear to be much morescattered if they were plotted on linear scales.

The maximum boundary 111 indicates the maximum dielectric constantexpected to be observed at each corresponding conductivity. In a similarfashion, the minimum boundary 115 represents the minimum dielectricconstant expected to be observed at each corresponding conductivity. Thepoints 122 through 128 represent measured values that fall somewhere inbetween the minimum boundary 115 and the maximum boundary 111. A medianline 113 is a line drawn so that half the points, or points 121 through124 are below the median line 113 and half the points, or points 126through 129 are above the median line 113. The point 125 falls right ontop of the median line 113.

An elemental measurement between a single transmitting 205 and a singlereceiving coil 207 is difficult to achieve in practice, but it is usefulfor describing the sensitivity of the measurement to variations of theconductivity and dielectric constant within a localized volume 225 of anearth formation 215. FIG. 2 illustrates in more detail specifically whatis meant by the term “sensitivity function,” also referred to as a“response function” or “geometrical factors.” Practitioners skilled inthe art of wireline logging, Measurement-While-Drilling (MWD) andLogging-While-Drilling (LWD) are familiar with how to generalize theconcept of a sensitivity function to apply to realistic measurementsfrom devices using multiple transmitting and receiving antennas.Typically a MWD/LWD resistivity measurement device transmits a signalusing a transmitter coil and measures the phase and magnitude of thesignal at one receiver antenna 307 relative to the values of the phaseand the magnitude at another receiver antenna 309 within a borehole 301(FIG. 3). These relative values are commonly referred to as the phaseshift and attenuation. It should be understood that one way to representa complex signal with multiple components is as a phasor signal.

Sensitivity Functions

FIG. 2 illustrates an exemplary single transmitter, single receiverMWD/LWD resistivity tool 220 for investigating an earth formation 215. Ametal shaft, or “mandrel,” 203 is incorporated within the drill string(the drill string is not shown, but it is a series of pipes screwedtogether with a drill bit on the end), inserted into the borehole 201,and employed to take measurements of an electrical signal thatoriginates at a transmitter 205 and is sensed at a receiver 207. Themeasurement tool is usually not removed from the well until the drillstring is removed for the purpose of changing drill bits or becausedrilling is completed. Selected data from the tool are telemetered tothe surface while drilling. All data are typically recorded in memorybanks for retrieval after the tool is removed from the borehole 201.Devices with a single transmitter and a single receiver are usually notused in practice, but they are helpful for developing concepts such asthat of the sensitivity function. Schematic drawings of simple,practical apparatuses are shown in FIGS. 3 and 4.

In a wireline operation, the measurement apparatus is connected to acable (known as a wireline), lowered into the borehole 201, and data areacquired. This is done typically after the drilling operation isfinished. Wireline induction tools measure the real (R) and imaginary(X) components of the receiver 207 signal. The R and X-signalscorrespond to the phase shift and attenuation measurements respectively.In order to correlate the sensitivity of the phase shift and attenuationmeasurements to variations in the conductivity and dielectric constantof the earth formation 215 at different positions within the earthformation, the conductivity and dielectric constant within a smallvolume P 225 are varied. For simplicity, the volume P 225 is a solid ofrevolution about the tool axis (such a volume is called atwo-dimensional volume). The amount the phase and attenuationmeasurements change relative to the amount the conductivity anddielectric constant changed within P 225 is essentially the sensitivity.The sensitivity function primarily depends on the location of the pointP 225 relative to the locations of the transmitter 205 and receiver 207,on the properties of the earth formation 215, and on the excitationfrequency. It also depends on other variables such as the diameter andcomposition of the mandrel 203, especially when P is near the surface ofthe mandrel 203.

Although the analysis is carried out in two-dimensions, the importantconclusions regarding the sensitive volumes of phase shift andattenuation measurements with respect to the conductivity and dielectricconstant hold in three-dimensions. Consequently, the scope of thisapplication is not limited to two-dimensional cases. This is discussedmore in a subsequent section entitled, “ITERATIVE FORWARD MODELING ANDDIPPING BEDS.”

The sensitivity function can be represented as a complex number having areal and an imaginary part. In the notation used below, S, denotes acomplex sensitivity function, and its real part is S′, and its imaginarypart is S″. Thus, S=S′+iS″, in which the imaginary number i={square rootover (−1)}. The quantities S′ and S″ are commonly referred to asgeometrical factors or response functions. The volume P 225 is located adistance ρ in the radial direction from the tool's axis and a distance zin the axial direction from the receiver 206. S′ represents thesensitivity of attenuation to resistivity and the sensitivity of phaseshift to dielectric constant. Likewise, S″ represents the sensitivity ofattenuation to dielectric constant and the sensitivity of phase shift toresistivity. The width of the volume P 225 is Δρ 211 and the height ofthe volume P 225 is Δz 213. The quantity S′, or the sensitivity ofattenuation to resistivity, is calculated by determining the effect achange in the conductivity (reciprocal of resistivity) in volume P 225from a prescribed background value has on the attenuation of a signalbetween the transmitter 205 and the receiver 207, assuming thebackground conductivity value is otherwise unperturbed within the entireearth formation 215. In a similar fashion, S″, or the sensitivity of thephase to the resistivity, is calculated by determining the effect achange in the conductivity value in the volume P 225 from an assumedbackground conductivity value has on the phase of the signal between thetransmitter 205 and the receiver 207, assuming the background parametersare otherwise unperturbed within the earth formation 215. Alternatively,one could determine S′ and S″ by determining the effect a change thedielectric constant within the volume P 225 has on the phase andattenuation, respectively. When the sensitivities are determined byconsidering a perturbation to the dielectric constant value within thevolume P 225, it is apparent that the sensitivity of the attenuation tochanges in the dielectric constant is the same as the sensitivity of thephase to the conductivity. It is also apparent that the sensitivity ofthe phase to the dielectric constant is the same as the sensitivity ofthe attenuation to the conductivity. By simultaneously considering thesensitivities of both the phase and attenuation measurement to thedielectric constant and to the conductivity, the Applicant shows apreviously unknown relationship between the attenuation and phase shiftmeasurements and the conductivity and dielectric constant values. Byemploying this previously unknown relationship, the Applicant providestechniques that produce better estimates of both the conductivity andthe dielectric constant values than was previously available from thosewith skill in the art. The sensitivity functions S′ and S″ and theirrelation to the subject matter of the Applicant's disclosure isexplained in more detail below in conjunction with FIGS. 5a-d and FIGS.6a-d.

In the above, sensitivities to the dielectric constant were referred to.Strictly speaking, the sensitivity to the radian frequency ω times thedielectric constant should have been referred to. This distinction istrivial to those skilled in the art.

In FIG. 2, if the background conductivity (reciprocal of resistivity) ofthe earth formation 215 isσ₀ and the background dielectric constant ofthe earth formation 215 is ∈₀, then the ratio of the receiver 207voltage to the transmitter 205 current in the background medium can beexpressed as Z_(RT) ⁰, where R stands for the receiver 207 and T standsfor the transmitter 205. Hereafter, a numbered subscript or superscriptsuch as the ‘0’ is merely used to identify a specific incidence of thecorresponding variable or function. If an exponent is used, the variableor function being raised to the power indicated by the exponent will besurrounded by parentheses and the exponent will be placed outside theparentheses. For example (L₁)³ would represent the variable L₁ raised tothe third power.

When the background conductivity σ₀ and/or dielectric constant ∈₀ arereplaced new values σ₁ and/or ∈₁ in the volume P 225, the ratio betweenthe receiver 207 voltage to the transmitter 205 current is representedby Z_(RT) ¹. Using the same nomenclature, a ratio between a voltage at ahypothetical receiver placed in the volume P 225 and the current at thetransmitter 205 can be expressed as Z_(PT) ⁰. In addition, a ratiobetween the voltage at the receiver 207 and a current at a hypotheticaltransmitter in the volume P 225 can be expressed as Z_(RP) ⁰. Using theBorn approximation, it can be shown that,$\frac{Z_{RT}^{1}}{Z_{RT}^{0}} = {1 + {{S\left( {T,R,P} \right)}\Delta \overset{\sim}{\sigma}{\Delta\rho\Delta}\quad z}}$

where the sensitivity function, defined as S(T,R,P), is${S\left( {T,R,P} \right)} = {- \frac{Z_{RP}^{0}Z_{PT}^{0}}{2{\pi\rho}\quad Z_{RT}^{0}}}$

in which Δ{tilde over (σ)}={tilde over (σ)}₁−{tilde over(σ)}₀=(σ₁−σ₀)+iω(∈₁−∈₀), and the radian frequency of the transmittercurrent is ω=2πƒ. A measurement of this type, in which there is just onetransmitter 205 and one receiver 207, is defined as an “elemental”measurement. It should be noted that the above result is also valid ifthe background medium parameters vary spatially within the earthformation 215. In the above equations, both the sensitivity functionS(T, R, P) and the perturbation Δ{tilde over (σ)} are complex-valued.Some disclosed embodiments consistently treat the measurements, theirsensitivities, and the parameters to be estimated as complex-valuedfunctions. This is not done in the prior art.

The above sensitivity function of the form S(T, R, P) is referred to asa 2-D (or two-dimensional) sensitivity function because the volume ΔρΔzsurrounding the point P 225, is a solid of revolution about the axis ofthe tool 201. Because the Born approximation was used, the sensitivityfunction S depends only on the properties of the background mediumbecause it is assumed that the same field is incident on the point P(ρ,z) even though the background parameters have been replaced by {tildeover (σ)}₁.

FIG. 3 illustrates an exemplary one-transmitter, two-receiver MWD/LWDresistivity measurement apparatus 320 for investigating an earthformation 315. Due to its configuration, the tool 320 is defined as an“uncompensated” device and collects uncompensated measurements from theearth formation 315. For the sake of simplicity, a borehole is notshown. This measurement tool 320 includes a transmitter 305 and tworeceivers 307 and 309, each of which is incorporated into a metalmandrel 303. Typically, the measurement made by such a device is theratio of the voltages at receivers 307 and 309. In this example, usingthe notation described above in conjunction with FIG. 2, the sensitivityfunction S(T, R, R′, P) for the uncompensated device can be shown to bethe difference between the elemental sensitivity functions S(T,R,P) andS(T, R′, P), where T represents the transmitter 305, R represents thereceiver 307, R′ represents the receiver 309, and P represents a volume(not shown) similar to the volume P 225 of FIG. 2.

For wireline induction measurements, the voltage at the receiver R issubtracted from the voltage at the receiver R′, and the position andnumber of turns of wire for R are commonly chosen so that thedifference, in the voltages at the two receiver antennas is zero whenthe tool is in a nonconductive medium., For MWD/LWD resistivity andwireline dielectric constant measurements, the voltage at the receiverR, or V_(R), and the voltage at the receiver R′, or V_(R′), are examinedas the ratio V_(R)/V_(R′). In either case, it can be shown that

S(T,R,R′,P)=S(T,R,P)−S(T,R′,P).

The sensitivity for an uncompensated measurement is the differencebetween the sensitivities of two elemental measurements such as S(T,R,P)and S(T,R′,P) calculated as described above in conjunction with FIG. 2.

FIG. 4 illustrates an exemplary two-transmitter, two-receiver MWD/LWDresistivity tool 420. Due to its configuration (transmitters beingdisposed symmetrically), the tool 420 is defined as a “compensated” tooland collects compensated measurements from an earth formation 415. Thetool 420 includes two transmitters 405 and 411 and two receivers 407 and409, each of which is incorporated into a metal mandrel or collar 403.Each compensated measurement is the geometric mean of two correspondinguncompensated measurements. In other words, during a particulartimeframe, the tool 420 performs two uncompensated measurements, oneemploying transmitter 405 and the receivers 407 and 409 and the otheremploying the transmitter 411 and the receivers 409 and 407. These twouncompensated measurements are similar to the uncompensated measurementdescribed above in conjunction with FIG. 3. The sensitivity function Sof the tool 420 is then defined as the arithmetic average of thesensitivity functions for each of the uncompensated measurements.Another way to describe this relationship is with the following formula:${S\left( {T,R,R^{\prime},T^{\prime},P} \right)} = {\frac{1}{2}\left\lbrack {{S\left( {T,R,R^{\prime},P} \right)} + {S\left( {T^{\prime},R^{\prime},R,P} \right)}} \right\rbrack}$

where T represents transmitter 405, T′ represents transmitter 411, Rrepresents receiver 407, R′ represents receiver 409 and P represents asmall volume of the earth formation similar to 225 (FIG. 2).

The techniques of the disclosed embodiments are explained in terms of acompensated tool such as the tool 420 and compensated measurements suchas those described in conjunction with FIG. 4. However, it should beunderstood that the techniques also apply to uncompensated tools such asthe tool 320 and uncompensated measurements described above inconjunction with FIG. 3 and elemental tools such as the tool 220 andelemental measurements such as those described above in conjunction withFIG. 2. In addition, the techniques are applicable for use in a wirelinesystem, a system that may not incorporate its transmitters and receiversinto a metal mandrel, but may rather affix a transmitter and a receiverto a tool made of a non-conducting material such as fiberglass. Thewireline induction frequency is typically too low for dielectric effectsto be significant. Also typical for wireline induction systems is toselect the position and number of turns of groups of receiver antennasso that there is a null signal in a nonconductive medium. When this isdone, Z_(RT) ⁰=0 if {tilde over (σ)}₀=0. As a result, it is necessary tomultiply the sensitivity and other quantities by Z_(RT) ⁰ to use theformulation given here in such cases.

The quantity Z_(RT) ¹/Z_(RT) ⁰ can be expressed as a complex numberwhich has both a magnitude and a phase (or alternatively real andimaginary parts). To a good approximation, the raw attenuation value(which corresponds to the magnitude) is:${{\frac{Z_{RT}^{1}}{Z_{RT}^{0}}} \approx {{Re}{\frac{Z_{RT}^{1}}{Z_{RT}^{0}}}}} = {{1 + {{{Re}\left\lbrack {{S\left( {T,R,P} \right)}\Delta \overset{\sim}{\sigma}} \right\rbrack}{\Delta\rho\Delta}\quad z}} = {1 + {\left\lbrack {{S^{\prime}{\Delta\sigma}} - {S^{''}{\omega\Delta ɛ}}} \right\rbrack {\Delta\rho\Delta}\quad z}}}$

where the function Re[•] denotes the real part of its argument. Also, toa good approximation, the raw phase shift value is:${{{phase}\left( \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right)} \approx {{Im}\left\lbrack \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \right\rbrack}} = {{{{Im}\left\lbrack {{S\left( {T,R,P} \right)}\Delta \overset{\sim}{\sigma}} \right\rbrack}{\Delta\rho\Delta}\quad z} = {\left\lbrack {{S^{''}{\Delta\sigma}} + {S^{\prime}{\omega\Delta ɛ}}} \right\rbrack {\Delta\rho\Delta}\quad z}}$

in which Im[•] denotes the imaginary part of its argument, S(T, R,P)=S′+iS″, Δσ=σ₁−σ₀, and Δ∈=∈₁−∈₀. For the attenuation measurement, S′is the sensitivity to the resistivity and S″ is the sensitivity to thedielectric constant. For the phase shift measurement, S′ is thesensitivity to the dielectric constant and S″ is the sensitivity to theresistivity. This is apparent because S′ is the coefficient of Δσ in theequation for attenuation, and it is also the coefficient of ωΔ∈ in theequation for the phase shift. Similarly, S″ is the coefficient of Δσ inthe equation for the phase shift, and it is also the coefficient for−ωΔ∈ in the equation for attenuation. This implies that the attenuationmeasurement senses the resistivity in the same volume as the phase shiftmeasurement senses the dielectric constant and that the phase shiftmeasurement senses the resistivity in the same volume as the attenuationmeasurement senses the dielectric constant. In the above, we havereferred to sensitivities to the dielectric constant. Strictly speaking,the sensitivity to the radian frequency ω times the dielectric constantΔ∈ should have been referred to. This distinction is trivial to thoseskilled in the art.

The above conclusion regarding the volumes in which phase andattenuation measurements sense the resistivity and dielectric constantfrom Applicant's derived equations also follows from a well known resultfrom complex variable theory known in that art as the Cauchy-Reimannequations. These equations provide the relationship between thederivatives of the real and imaginary parts of an analytic complexfunction with respect to the real and imaginary parts of the function'sargument.

FIGS. 5a, 5 b, 5 c and 5 d can best be described and understoodtogether. In all cases, the mandrel diameter is 6.75 inches, thetransmitter frequency is 2 MHz, and the background medium ischaracterized by a conductivity of σ₀=0.01S/m and a relative dielectricconstant of ∈₀=10. The data in FIGS. 5a and 5 c labeled “MediumMeasurement” are for a compensated type of design shown in FIG. 4. Theexemplary distances between transmitter 405 and receivers 407 and 409are 20 and 30 inches, respectively. Since the tool is symmetric, thedistances between transmitter 411 and receivers 409 and 407 are 20 and30 inches, respectively. The data in FIGS. 5b and 5 d labeled “DeepMeasurement” are also for a compensated tool as shown in FIG. 4, butwith exemplary transmitter-receiver spacings of 50 and 60 inches. Eachplot shows the sensitivity of a given measurement as a function ofposition within the formation. The term sensitive volume refers to theshape of each plot as well as its value at any point in the formation.The axes labeled “Axial Distance” refer to the coordinate along the axisof the tool with zero being the geometric mid-point of the antenna array(halfway between receivers 407 and 409) to a given point in theformation. The axes labeled “Radial Distance” refer to the radialdistance from the axis of the tool to a given point in the formation.The value on the vertical axis is actually the sensitivity value for theindicated measurement. Thus, FIG. 5a is a plot of a sensitivity functionthat illustrates the sensitivity of the “Medium” phase shift measurementin relation to changes in the resistivity as a function of the locationof the point P 225 in the earth formation 215 (FIG. 2). If themeasurement of phase shift changes significantly in response to changingthe resistivity from its background value, then phase shift isconsidered relatively sensitive to the resistivity at the point P 225.If the measurement of phase shift does not change significantly inresponse to changing the resistivity, then the phase shift is consideredrelatively insensitive at the point P 225. Based upon the relationshipdisclosed herein, FIG. 5a also illustrates the sensitivity of the“Medium” attenuation measurement in relation to changes in dielectricconstant values. Note that the dimensions of the sensitivity on thevertical axes is ohms per meter (Ω/m) and distances on the horizontalaxes are listed in inches. In a similar fashion, FIG. 5b is a plot ofthe sensitivity of the attenuation measurement to the resistivity. Basedon the relationship disclosed herein, FIG. 5b is also the sensitivity ofa phase shift measurement to a change in the dielectric constant. FIGS.5b and 5 d have the same descriptions as FIGS. 5a and 5 c, respectively,but FIGS. 5b and 5 d are for the “Deep Measurement” with the antennaspacings described above.

Note that the shape of FIG. 5a is very dissimilar to the shape of FIG.5c. This means that the underlying measurements are sensitive to thevariables in different volumes. For example, the Medium phase shiftmeasurement has a sensitive volume characterized by FIG. 5a for theresistivity, but this measurement has the sensitive volume shown in FIG.5c for the dielectric constant. As discussed below, it is possible totransform an attenuation and a phase shift measurement to a complexnumber which has the following desirable properties: 1) its real part issensitive to the resistivity in the same volume that the imaginary partis sensitive to the dielectric constant; 2) the real part has no netsensitivity to the dielectric constant; and, 3) the imaginary part hasno net sensitivity to the resistivity. In addition, the transformationis generalized to accommodate multiple measurements acquired at multipledepths. The generalized method can be used to produce independentestimates of the resistivity and dielectric constant within a pluralityof volumes within the earth formation.

Transformed Sensitivity Functions and Transformation of the Measurements

For simplicity, the phase shift and attenuation will not be used.Hereafter, the real and imaginary parts of measurement will be referredto instead. Thus,

w=w′+iw″

w′=(10)^(db/20)×cos(θ)

w″=(10)^(dB/20)×sin(θ)

where w′ is the real part of w, w″ is the imaginary part of w, i is thesquare root of the integer −1, dB is the attenuation in decibels, and θis the phase shift in radians.

The equations that follow can be related to the sensitivity functionsdescribed above in conjunction with FIG. 2 by defining variablesw₁=Z_(RT) ¹ and w₀=Z_(RT) ⁰. The variable w₁ denotes an actual toolmeasurement in the earth formation 215. The variable w₀ denotes theexpected value for the tool measurement in the background earthformation 215. For realistic measurement devices such as those describedin FIGS. 3 and 4, the values for w₁ and w₀ would be the voltage ratiosdefined in the detailed description of FIGS. 3 and 4. In one embodiment,the parameters for the background medium are determined and then used tocalculate value of w₀ using a mathematical model to evaluate the toolresponse in the background medium. One of many alternative methods todetermine the background medium parameters is to estimate w₀ directlyfrom the measurements, and then to determine the background parametersby correlating w₀ to a model of the tool in the formation which has thebackground parameters as inputs.

As explained in conjunction with FIG. 2, the sensitivity function Srelates the change in the measurement to a change in the mediumparameters such as resistivity and dielectric constant within a smallvolume 225 of the earth formation 215 at a prescribed location in theearth formation 225, or background medium. A change in measurements dueto small variations in the medium parameters at a range of locations canbe calculated by integrating the responses from each such volume in theearth formation 215. Thus, if Δ{tilde over (σ)} is defined for a largenumber of points ρ,z, then

Z _(RT) ¹ =Z _(RT) ⁰(1+I[SΔ{tilde over (σ)}])

in which I is a spatial integral function further defined as${\Delta \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{w_{1}}{w_{0}} - 1} \right)\quad {and}}}}$$\hat{S} = \frac{S}{I\lbrack S\rbrack}$

where F is a complex function.

Although the perturbation from the background medium, Δ{tilde over (σ)}is a function of position, parameters of a hypothetical, equivalenthomogeneous perturbation (meaning that no spatial variations are assumedin the difference between the resistivity and dielectric constant andvalues for both of these parameters in the background medium) can bedetermined by assuming the perturbation is not a function of positionand then solving for it. Thus,

Δ{circumflex over (σ)}I[S]=I[SΔ{tilde over (σ)}]

where Δ{circumflex over (σ)} represents the parameters of the equivalenthomogeneous perturbation. From the previous equations, it is clear thatI[F] = ∫_(−∞)^(+∞)  z∫₀^(+∞)  ρ  F(ρ, z)

where Δ{circumflex over (σ)} is the transformed measurement (it isunderstood that Δ{circumflex over (σ)} is also the equivalenthomogeneous perturbation and that the terms transformed measurement andequivalent homogeneous perturbation will be used synonymously), Ŝ is thesensitivity function for the transformed measurement, and Ŝ will bereferred to as the transformed sensitivity function. In the above, w₁ isthe actual measurement, and w₀ is the value assumed by the measurementin the background medium. An analysis of the transformed sensitivityfunction Ŝ, shows that the transformed measurements have the followingproperties: 1) the real part of Δ{circumflex over (σ)} is sensitive tothe resistivity in the same volume that its imaginary part is sensitiveto the dielectric constant; 2) the real part of Δ{circumflex over (σ)}has no net sensitivity to the dielectric constant; and, 3) the imaginarypart of Δ{circumflex over (σ)} has no net sensitivity to theresistivity. Details of this analysis will be given in the next fewparagraphs.

The techniques of the disclosed embodiment can be further refined byintroducing a calibration factor c (which is generally a complex numberthat may depend on the temperature of the measurement apparatus andother environmental variables) to adjust for anomalies in the physicalmeasurement apparatus. In addition, the term, w_(bh) can be introducedto adjust for effects caused by the borehole 201 on the measurement.With these modifications, the transformation equation becomes${\Delta \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}{\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right).}}}}$

The sensitivity function for the transformed measurement is determinedby applying the transformation to the original sensitivity function, S.Thus,$\hat{S} = {{{\hat{S}}^{\prime} + {i{\hat{S}}^{''}}} = {\frac{S}{I\lbrack S\rbrack} = {\frac{{S^{\prime}{I\left\lbrack S^{\prime} \right\rbrack}} + {S^{''}{I\left\lbrack S^{''} \right\rbrack}}}{{{I\lbrack S\rbrack}}^{2}} + {{\frac{{S^{''}{I\left\lbrack S^{\prime} \right\rbrack}} - {S^{\prime}{I\left\lbrack S^{''} \right\rbrack}}}{{{I\lbrack S\rbrack}}^{2}}.}}}}}$

Note that I[Ŝ]=I[Ŝ′]=1 because I[Ŝ″]=0. The parameters for theequivalent homogeneous perturbation are

Δ{circumflex over (σ)}′={circumflex over (σ)}₁−σ₀ =I[Ŝ′Δσ]−I[Ŝ″ωΔ∈

Δ{circumflex over (σ)}″=ω({circumflex over (∈)}₁−∈₀)=I[Ŝ′ωΔ∈]+I[Ŝ″Δσ]

The estimate for the conductivity perturbation, Δ{circumflex over (σ)}′suppresses sensitivity (is relatively insensitive) to the dielectricconstant perturbation, and the estimate of the dielectric constantperturbation, Δ{circumflex over (σ)}″/ω suppresses sensitivity to theconductivity perturbation. This is apparent because the coefficient ofthe suppressed variable is Ŝ″. In fact, the estimate for theconductivity perturbation Δ{circumflex over (σ)}′ is independent of thedielectric constant perturbation provided that deviations in thedielectric constant from its background are such that I[Ŝ″ωΔ∈]=0. SinceI[Ŝ″]=0, this is apparently the case if ωΔ∈ is independent of position.Likewise, the estimate for the dielectric constant perturbation given byΔ{circumflex over (σ)}″/ω is independent of the conductivityperturbation provided that deviations in the conductivity from itsbackground value are such that I[Ŝ″Δσ]=0. Since I[Ŝ″]=0, this isapparently the case if Δσ is independent of position.

Turning now to FIGS. 6a and 6 b, illustrated are plots of thesensitivity functions Ŝ′ and Ŝ″ derived from S′ and S″ for the mediumtransmitter-receiver spacing measurement shown in FIGS. 5a and 5 c usingthe transformation $\hat{S} = {\frac{S}{I\lbrack S\rbrack}.}$

The data in FIGS. 6c and 6 d were derived from the data in FIGS. 5b and5 d for the Deep T-R spacing measurement. As shown in FIGS. 6a, 6 b, 6 cand 6 d, using the transformed measurements to determine the electricalparameters of the earth formation is a substantial improvement over theprior art. The estimates of the medium parameters are more accurate andless susceptible to errors in the estimate of the background mediumbecause the calculation of the resistivity is relatively unaffected bythe dielectric constant and the calculation of the dielectric constantis relatively unaffected by the resistivity. In addition to integratingto 0, the peak values for Ŝ″ in FIGS. 6b and 6 d are significantly lessthan the respective peak values for Ŝ′ in FIGS. 6a and 6 c. Both ofthese properties are very desirable because Ŝ″ is the sensitivityfunction for the variable that is suppressed.

Realization of the Transformation

In order to realize the transformation, it is desirable to have valuesof I[S] readily accessible over the range of background mediumparameters that will be encountered. One way to achieve this is tocompute the values for I[S] and then store them in a lookup table foruse later. Of course, it is not necessary to store these data in such alookup table if it is practical to quickly calculate the values for I[S]on command when they are needed. In general, the values for I[S] can becomputed by directly; however, it can be shown that${{{I\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}$

where w₀ is the expected value for the measurement in the backgroundmedium, and the indicated derivative is calculated using the followingdefinition:${\frac{\partial w}{\partial\overset{\sim}{\sigma}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\lim\limits_{{\Delta \overset{\sim}{\sigma}}\rightarrow 0}{\cdot {\frac{{w\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta \overset{\sim}{\sigma}}} \right)} - {w\left( {\overset{\sim}{\sigma}}_{0} \right)}}{\left( {{\overset{\sim}{\sigma}}_{0} + {\Delta \overset{\sim}{\sigma}}} \right) - {\overset{\sim}{\sigma}}_{0}}.}}}$

In the above formula, {tilde over (σ)}₀ may vary from point to point inthe formation 215 (the background medium may be inhomogeneous), but theperturbation Δ{tilde over (σ)} is constant at all points in theformation 215. As an example of evaluating I[S] using the above formula,consider the idealized case of a homogeneous medium with a smalltransmitter coil and two receiver coils spaced a distance L₁ and L₂ fromthe transmitter. Then,${{{w_{0} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp \left( {\quad k_{0}L_{2}} \right)}\left( {1 - {\quad k_{0}L_{2}}} \right)}{{\exp \left( {\quad k_{0}L_{1}} \right)}\left( {1 - {\quad k_{0}L_{1}}} \right)}}}{{I\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\frac{\omega\mu}{2}\left( {\frac{\left( L_{2} \right)^{2}}{1 - {\quad k_{0}L_{2}}} - \frac{\left( L_{1} \right)^{2}}{1 - {\quad k_{0}L_{1}}}} \right)}$

The wave number in the background medium is k₀={square root over(iωμ{tilde over (σ)})}₀, the function exp(•) is the complex exponentialfunction where exp(1)≈2.71828, and the symbol μ denotes the magneticpermeability of the earth formation. The above formula for I[S] appliesto both uncompensated (FIG. 3) and to compensated (FIG. 4) measurementsbecause the background medium has reflection symmetry about the centerof the antenna array in FIG. 4.

For the purpose of this example, the above formula is used to computethe values for I[S]=I[S′]+iI[S″]. FIG. 7 illustrates an exemplary table701 employed in a Create Lookup Table step 903 (FIG. 9) of the techniqueof the disclosed embodiment. Step 903 generates a table such as table701 including values for the integral of the sensitivity function overthe range of variables of interest. The first two columns of the table701 represent the conductivity σ₀ and the dielectric constant ∈₀ of thebackground medium. The third and fourth columns of the table 701represent calculated values for the functions I[S′] and I[S″] for a Deepmeasurement, in which the spacing between the transmitter 305 receivers307 and 309 is 50 and 60 inches, respectively. The fifth and sixthcolumns of the table 701 represent calculated values for the functionsI[S′] and I[S″] for a Medium measurement, in which the spacing betweenthe transmitter 305 receivers 307 and 309 is 20 and 30 inches,respectively. It is understood that both the frequency of thetransmitter(s) and the spacing between the transmitter(s) andreceiver(s) can be varied. Based upon this disclosure, it is readilyapparent to those skilled in the art that algorithms such as the onedescribed above can be applied to alternative measurementconfigurations. If more complicated background media are used, forexample including the mandrel with finite-diameter antennas, it may bemore practical to form a large lookup table such as table 701 but withmany more values. Instead of calculating I[S] every time a value isneeded, data would be interpolated from the table. Nonetheless, table701 clearly illustrates the nature of such a lookup table. Such a tablewould contain the values of the functions I[S′] and I[S″] for the entirerange of values of the conductivity σ₀ and the dielectric constant ∈₀likely to be encountered in typical earth formations. For example, I[S′]and I[S″] could be calculated for values of ∈₀ between 1 and 1000 andfor values of σ₀ between 0.0001 and 10.0. Whether calculating values forthe entire lookup table 701 or computing the I[S′] and I[S″] on commandas needed, the data is used as explained below.

FIG. 8 illustrates a chart 801 used to implement a Determine BackgroundMedium Parameters step 905 (FIG. 9) of the techniques of the disclosedembodiment. The chart 801 represents a plot of the attenuation and phaseshift as a function of resistivity and dielectric constant in ahomogeneous medium. Similar plots can be derived for more complicatedmedia. However, the homogeneous background media are routinely used dueto their simplicity. Well known numerical methods such as inverseinterpolation can be used to calculate an initial estimate of backgroundparameters based upon the chart 801. In one embodiment, the measuredattenuation and phase shift values are averaged over a few feet of depthwithin the borehole 201. These average values are used to determine thebackground resistivity and dielectric constant based upon the chart 801.It should be understood that background medium parameters can beestimated in a variety of ways using one or more attenuation and phasemeasurements.

FIG. 9 is a flowchart of an embodiment of the disclosed transformationtechniques that can be implemented in a software program which isexecuted by a processor of a computing system such as a computer at thesurface or a “downhole” microprocessor. Starting in a Begin Analysisstep 901, control proceeds immediately to the Create Lookup Table step903 described above in conjunction with FIG. 7. In an alternativeembodiment, step 903 can be bypassed and the function of the lookuptable replaced by curve matching, or “forward modeling.” Control thenproceeds to an Acquire Measured Data Step 904. Next, control proceeds toa Determine Background Values Step 905, in which the background valuesfor the background medium are determined. Step 905 corresponds to thechart 801 (FIG. 8).

Control then proceeds to a Determine Integral Value step 907. TheDetermine Integral Value step 907 of the disclosed embodiment determinesan appropriate value for I[S] using the lookup table generated in thestep 903 described above or by directly calculating the I[S] value asdescribed in conjunction with FIG. 7. Compute Parameter Estimate, step909, computes an estimate for the conductivity and dielectric constantas described above using the following equation:${\Delta \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}{\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right).}}}}$

where the borehole effect and a calibration factor are taken intoaccount using the factors w_(bh) and c, respectively. The conductivityvalue plotted on the log (this is the value correlated to theconductivity of the actual earth formation) is Re(Δ{circumflex over(σ)}+{tilde over (σ)}₀) where the background medium is characterized by{tilde over (σ)}₀. The estimate for the dielectric constant can also beplotted on the log (this value is correlated to the dielectric constantof the earth formation), and this value is Im(Δ{circumflex over(σ)}+{tilde over (σ)}₀)/ω. Lastly, in the Final Depth step 911, it isdetermined whether the tool 201 is at the final depth within the earthformation 215 that will be considered in the current logging pass. Ifthe answer is “Yes,” then control proceeds to a step 921 where isprocessing is complete. If the answer in step 911 is “No,” controlproceeds to a Increment Depth step 913 where the tool 220 is moved toits next position in the borehole 201 which penetrates the earthformation 215. After incrementing the depth of the tool 220, controlproceeds to step 904 where the process of steps 904, 905, 907, 909 and911 are repeated. It should be understood by those skilled in the artthat embodiments described herein in the form a computing system or as aprogrammed electrical circuit can be realized.

Improved estimates for the conductivity and/or dielectric constant canbe determined by simultaneously considering multiple measurements atmultiple depths. This procedure is described in more detail below underthe heading “Multiple Sensors At Multiple Depths.”

Multiple Sensors at Multiple Depths

In the embodiments described above, the simplifying assumption thatΔ{tilde over (σ)} is not position dependent facilitates determining avalue for Δ{circumflex over (σ)} associated with each measurement byconsidering only that measurement at a single depth within the well (atleast given a background value {tilde over (σ)}₀). It is possible toeliminate the assumption that Δ{tilde over (σ)} is independent ofposition by considering data at multiple depths, and in general, to alsoconsider multiple measurements at each depth. An embodiment of such atechnique for jointly transforming data from multiple MWD/LWD sensors atmultiple depths is given below. Such an embodiment can also be used forprocessing data from a wireline dielectric tool or a wireline inductiontool. Alternate embodiments can be developed based on the teachings ofthis disclosure by those skilled in the art.

In the disclosed example, the background medium is not assumed to be thesame at all depths within the processing window. In cases where it ispossible to assume the background medium is the same at all depthswithin the processing window, the system of equations to be solved is inthe form of a convolution. The solution to such systems of equations canbe expressed as a weighted sum of the measurements, and the weights canbe determined using standard numerical methods. Such means are known tothose skilled in the art, and are referred to as “deconvolution”techniques. It will be readily understood by those with skill in the artthat deconvolution techniques can be practiced in conjunction with thedisclosed embodiments without departing from the spirit of theinvention, but that the attendant assumptions are not necessary topractice the disclosed embodiments in general.

Devices operating at multiple frequencies are considered below, butmultifrequency operation is not necessary to practice the disclosedembodiments. Due to frequency dispersion (i.e., frequency dependence ofthe dielectric constant and/or the conductivity value), it is notnecessarily preferable to operate using multiple frequencies. Given thedisclosed embodiments, it is actually possible to determine thedielectric constant and resistivity from single-frequency data. In fact,the disclosed embodiments can be used to determine and quantifydispersion by separately processing data sets acquired at differentfrequencies. In the below discussion, it is understood that subsets ofdata from a given measurement apparatus or even from several apparatusescan be processed independently to determine parameters of interest. Thebelow disclosed embodiment is based on using all the data availablestrictly for purpose of simplifying the discussion.

Suppose multiple transmitter-receiver spacings are used and that eachtransmitter is excited using one or more frequencies. Further, supposedata are collected at multiple depths in the earth formation 215. Let Ndenote the number of independent measurements performed at each ofseveral depths, where a measurement is defined as the data acquired at aparticular frequency from a particular set of transmitters and receiversas shown in FIGS. 3 or 4. Then, at each depth z_(k), a vector of all themeasurements can be defined as${{\overset{\_}{v}}_{k} = \left\lbrack {\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{k1},\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{k2},\cdots \quad,\left( {\frac{w_{1}}{w_{0}} - 1} \right)_{kN}} \right\rbrack^{T}},$

and the perturbation of the medium parameters from the background mediumvalues associated with these measurements is

Δ{tilde over ({overscore (σ)})}=Δ{tilde over (σ)}(ρ,z)[1,1, . . .,1]^(T)

in which the superscript T denotes a matrix transpose, {overscore(v)}_(k) is a vector each element of which is a measurement, and Δ{tildeover ({overscore (σ)})} is a vector each element of which is aperturbation from the background medium associated with a correspondingelement of {overscore (v)}_(k) at the point P 225. In the above, thedependence of the perturbation, Δ{tilde over (σ)}(ρ,z)on the position ofthe point P 225 is explicitly denoted by the variables ρ and z. Ingeneral, the conductivity and dielectric constant of both the backgroundmedium and the perturbed medium depend on ρ and z; consequently, nosubscript k needs to be associated with Δ{tilde over (σ)}(ρ,z), and allelements of the vector Δ{tilde over ({overscore (σ)})} are equal. Asdescribed above, borehole corrections and a calibration can be appliedto each measurement, but here they are omitted for simplicity.

The vectors {overscore (v)}_(k) and Δ{tilde over ({overscore (σ)})} arerelated as follows:

{overscore (v)} _(k) =I[{overscore ({overscore (S)})}Δ{tilde over({overscore (σ)})}]

in which {overscore ({overscore (S)})} is a diagonal matrix with eachdiagonal element being the sensitivity function centered on the depthz_(k), for the corresponding element of {overscore (v)}_(k), and theintegral operator I is defined by:I[F] = ∫_(−∞)^(+∞)  z∫₀^(+∞)  ρ  F(ρ, z).

Using the notationI_(mn)[F] = ∫_(z_(m − 1))^(z_(m))  z∫_(ρ_(n − 1))^(ρ_(n))  ρ  F(ρ, z)

to denote integrals of a function over the indicated limits ofintegration, it is apparent that${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}\quad {\sum\limits_{n = 1}^{N^{\prime}}\quad {I_{mn}\left\lbrack {\overset{\_}{\overset{\_}{S}}\Delta \overset{\_}{\overset{\sim}{\sigma}}} \right\rbrack}}}$

if ρ₀=0, ρ_(N′)=+∞, z_(−M−1)=−∞, and z_(M)=+∞. The equation directlyabove is an integral equation from which an estimate of Δ{tilde over(σ)}(ρ, z) can be calculated. With the definitionsρ_(n)*=(ρ_(n)+ρ_(n−1))/2 and Z_(m)*=(z_(m)+z_(m−1))/2 and making theapproximation Δ{tilde over (σ)}(ρ,z)=Δ{circumflex over(σ)}(ρ_(n)*,z_(m)*) within the volumes associated with each value for mand n, it follows that${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}\quad {\sum\limits_{n = 1}^{N^{\prime}}\quad {{I_{mn}\left\lbrack \overset{\_}{\overset{\_}{S}} \right\rbrack}\Delta {\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}}}}$

where N′≦N to ensure this system of equations is not underdetermined.The unknown values Δ{circumflex over ({overscore (σ)})}(ρ_(n)*, z_(m)*)can then be determined by solving the above set of linear equations. Itis apparent that the embodiment described in the section entitled“REALIZATION OF THE TRANSFORMATION” is a special case of the above forwhich M=0,N=N′=1.

Although the approximation Δ{tilde over (σ)}(ρ,z)=Δ{circumflex over(σ)}(ρ_(n)*, z_(m)*) (which merely states that Δ{tilde over (σ)}(ρ,z) isa piecewise constant function of ρ,z) is used in the immediately aboveembodiment, such an approximation is not necessary. More generally, itis possible to expand Δ{tilde over (σ)}(ρ, z) using a set of basisfunctions, and to then solve the ensuing set of equations for thecoefficients of the expansion. Specifically, suppose${{\Delta {\overset{\sim}{\sigma}\left( {\rho,z} \right)}} = {\sum\limits_{m = {- \infty}}^{\infty}\quad {\sum\limits_{n = {- \infty}}^{\infty}\quad {a_{mn}{\varphi_{mn}\left( {\rho,z} \right)}\quad {then}}}}},{{\overset{\_}{v}}_{k} = {\sum\limits_{m = {- \infty}}^{\infty}\quad {\sum\limits_{n = {- \infty}}^{\infty}\quad {{I\left\lbrack {\overset{\_}{\overset{\_}{S}}\varphi_{mn}} \right\rbrack}{\overset{\_}{a}}_{mn}}}}}$

where {overscore (a)}_(mn)=a_(mn)[1,1, . . . , 1]^(T). Some desirableproperties for the basis functions φ_(mn) are: 1) the integralsI[{overscore ({overscore (S)})}φ_(mn)] in the above equation all exist;and, 2) the system of equations for the coefficients a_(mn) is notsingular. It is helpful to select the basis functions so that a minimalnumber of terms is needed to form an accurate approximation to Δ{tildeover (σ)}(ρ,z).

The above embodiment is a special case for which the basis functions areunit step functions. In fact, employing the expansion${\Delta {\overset{\sim}{\sigma}\left( {\rho,z} \right)}} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{\Delta \quad {{{\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}\left\lbrack {{u\left( {z - z_{m}} \right)} - {u\left( {z - z_{m - 1}} \right)}} \right\rbrack}\left\lbrack {{u\left( {\rho - \rho_{n}} \right)} - {u\left( {\rho - \rho_{n - 1}} \right)}} \right\rbrack}}}}$

where u(•) denotes the unit step function leads directly to the samesystem of equations${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{{I_{m\quad n}\left\lbrack \overset{\overset{\_}{\_}}{S} \right\rbrack}\Delta \quad {\overset{\_}{\hat{\sigma}}\left( {\rho_{n}^{*},z_{m}^{*}} \right)}}}}$

given in the above embodiment. Specific values for M, N′, z_(m), andρ_(n) needed to realize this embodiment of the invention depend on theexcitation frequency(ies), on the transmitter-receiver spacings that areunder consideration, and generally on the background conductivity anddielectric constant. Different values for z_(m) and ρ_(n) are generallyused for different depth intervals within the same well because thebackground medium parameters vary as a function of depth in the well.

Solving the immediately above system of equations results in estimatesof the average conductivity and dielectric constant within the volume ofthe earth formation 215 corresponding to each integral I_(mn)[{overscore({overscore (S)})}]. In an embodiment, the Least Mean Square method isused to determine values for Δ{circumflex over ({overscore(σ)})}(ρ_(n)*z_(m)*) by solving the above system of equations. Manytexts on linear algebra list other techniques that may also be used.

Unlike other procedures previously used for processing MWD/LWD data, thetechniques of a disclosed embodiment account for dielectric effects andprovide for radial inhomogeneities in addition to bedding interfaces byconsistently treating the signal as a complex-valued function of theconductivity and the dielectric constant. This procedure producesestimates of one variable (i.e., the conductivity) are not corrupted byeffects of the other (i.e., the dielectric constant). As mentioned inthe above “SUMMARY OF THE INVENTION,” this result was deemedimpracticable as a consequence of the “old assumptions.”

A series of steps, similar to those of FIG. 9, can be employed in orderto implement the embodiment for Multiple Sensors at Multiple Depths.Since the lookup table for I_(mn)[{overscore ({overscore (S)})}] neededto realize such an embodiment could be extremely large, these values areevaluated as needed in this embodiment. This can be done in a manneranalogous to the means described in the above section “REALIZATION OFTHE TRANSFORMATION” using the following formulae:${I_{m\quad n}\lbrack S\rbrack} = {\left. {\frac{1}{w_{0}}\quad \frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{m\quad n}}} \middle| {}_{\overset{\sim}{\sigma}\quad = \quad {\overset{\sim}{\sigma}}_{0}}\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{m\quad n}} \right|_{\overset{\sim}{\sigma}\quad = \quad {\overset{\sim}{\sigma}}_{0}} = {\lim\limits_{{{\Delta \quad {\quad \overset{\sim}{\sigma}}_{m\quad n}}\quad\rightarrow\quad 0}\quad}\quad {\frac{{{w\left( \quad {{\overset{\sim}{\sigma}}_{0}\quad + \quad {\Delta \quad {\quad \overset{\sim}{\sigma}}_{m\quad n}}} \right)}\quad - \quad {w\left( {\overset{\sim}{\sigma}}_{0} \right)}}\quad}{\left( {{\overset{\sim}{\sigma}}_{0}\quad + \quad {\Delta \quad {\quad \overset{\sim}{\sigma}}_{m\quad n}}} \right)\quad - \quad {\overset{\sim}{\sigma}}_{0}}.}}}$

where {tilde over (σ)}_(mn)=σ_(mn)+iω∈_(mn) represents the conductivityand dielectric constant of the region of space over which the integralI_(mn)[S] is evaluated. In words, I_(mn)[S] can be calculated byevaluating the derivative of the measurement with respect to the mediumparameters within the volume covered by the integration. Alternatively,one could evaluate I_(mn)[S] by directly carrying out the integration asneeded. This eliminates the need to store the values in a lookup table.

While the above exemplary systems are described in the context of anMWD/LWD system, it shall be understood that a system according to thedescribed techniques can be implemented in a variety of other loggingsystems such as wireline induction or wireline dielectric measurementsystems. Further in accordance with the disclosed techniques, it shouldbe understood that phase shift and attenuation can be combined in avariety of ways to produce a component sensitive to resistivity andrelatively insensitive to dielectric constant and a component sensitiveto dielectric constant and relatively insensitive to resistivity. In theinstance of MWD/LWD resistivity measurement systems, resistivity is thevariable of primary interest; as a result, phase shift and attenuationmeasurements can be combined to produce a component sensitive toresistivity and relatively insensitive dielectric constant.

Single Measurements at a Single Depth

One useful embodiment is to correlate (or alternatively equate) a singlemeasured value w₁ to a model that predicts the value of the measurementas a function of the conductivity and dielectric constant within aprescribed region of the earth formation. The value for the dielectricconstant and conductivity that provides an acceptable correlation (oralternatively solves the equation) is then used as the final result(i.e., correlated to the parameters of the earth formation). Thisprocedure can be performed mathematically, or graphically. Plotting apoint on a chart such as FIG. 8 and then determining which dielectricvalue and conductivity correspond to it is an example of performing theprocedure graphically. It can be concluded from the preceding sections,that Ŝ is the sensitivity of such an estimate of the dielectric constantand conductivity to perturbations in either variable. Thus such aprocedure results in an estimate for the conductivity that has no netsensitivity to the changes in the dielectric constant and an estimatefor the dielectric constant that has no net sensitivity to changes inthe conductivity within the volume in question. This is a very desirableproperty for the results to have. The utility of employing a singlemeasurement at a single depth derives from the fact that data processingalgorithms using minimal data as inputs tend to provide results quicklyand reliably. This procedure is a novel means of determining of oneparameter (either the conductivity or the dielectric constant) with nonet sensitivity to the other parameter. Under the old assumptions, thisprocedure would appear to not be useful for determining independentparameter estimates.

Iterative Forward Modeling and Dipping Beds

The analysis presented above has been carried out assuming a2-dimensional geometry where the volume P 225 in FIG. 2 is a solid ofrevolution about the axis of the tool. In MWD/LWD and wirelineoperations, there are many applications where such a 2-dimensionalgeometry is inappropriate. For example, the axis of the tool oftenintersects boundaries between different geological strata at an obliqueangle. Practitioners refer to the angle between the tool axis and avector normal to the strata as the relative dip angle. When the relativedip angle is not zero, the problem is no longer 2-dimensional. However,the conclusion that: 1) the attenuation measurement is sensitive to theconductivity in the same volume as the phase measurement is sensitive tothe dielectric constant; and, 2) an attenuation measurement is sensitiveto the dielectric constant in the same volume that the phase measurementis sensitive to the conductivity remains true in the more complicatedgeometry. Mathematically, this conclusion follows from theCauchy-Reimann equations which still apply in the more complicatedgeometry (see the section entitled “SENSITIVITY FUNCTIONS”). Thephysical basis for this conclusion is that the conduction currents arein quadrature (90 degrees out of phase) with the displacement currents.At any point in the formation, the conduction currents are proportionalto the conductivity and the displacement currents are proportional tothe dielectric constant.

A common technique for interpreting MWD/LWD and wireline data inenvironments with complicated geometry such as dipping beds is to employa model which computes estimates for the measurements as a function ofthe parameters of a hypothetical earth formation. Once model inputparameters have been selected that result in a reasonable correlationbetween the measured data and the model data over a given depthinterval, the model input parameters are then correlated to the actualformation parameters. This process is often referred to as “iterativeforward modeling” or as “Curve Matching,” and applying it in conjunctionwith the old assumptions, leads to errors because the volumes in whicheach measurement senses each variable have to be known in order toadjust the model parameters appropriately.

The algorithms discussed in the previous sections can also be adaptedfor application to data acquired at non-zero relative dip angles.Selecting the background medium to be a sequence of layers having theappropriate relative dip angle is one method for so doing.

Transformations For a Resistivity-Dependent Dielectric Constant

In the embodiments described above, both the dielectric constant andconductivity are treated as independent quantities and the intent is toestimate one parameter with minimal sensitivity to the other. As shownin FIG. 1, there is empirical evidence that the dielectric constant andthe conductivity can be correlated. Such empirical relationships arewidely used in MWD/LWD applications, and when they hold, one parametercan be estimated if the other parameter is known.

This patent application shows that: 1) an attenuation measurement issensitive to the conductivity in the same volume of an earth formationas the phase measurement is sensitive to the dielectric constant; and,2) the attenuation measurement is sensitive to the dielectric constantin the same volume that the phase measurement is sensitive to theconductivity. A consequence of these relationships is that it is notgenerally possible to derive independent estimates of the conductivityfrom a phase and an attenuation measurement even if the dielectricconstant is assumed to vary in a prescribed manner as a function of theconductivity. The phrase “not generally possible” is used above becauseindependent estimates from each measurement can be still be made if thedielectric constant doesn't depend on the conductivity or if theconductivity and dielectric constant of earth formation are practicallythe same at all points within the sensitive volumes of bothmeasurements. Such conditions represent special cases which are notrepresentative of conditions typically observed within earth formations.

Even though two independent estimates of the conductivity are notgenerally possible from a single phase and a single attenuationmeasurement, it is still possible to derive two estimates of theconductivity from a phase and an attenuation measurement given atransformation to convert the dielectric constant into a variable thatdepends on the resistivity. For simplicity, consider a device such asthat of FIG. 3. Let the complex number w₁ denote an actual measurement(i.e., the ratio of the voltage at receiver 307 relative the voltage atreceiver 309, both voltages induced by current flowing throughtransmitter 305). Let the complex number w denote the value of saidmeasurement predicted by a model of the tool 320 in a prescribed earthformation 315. For further simplicity, suppose the model is as describedabove in the section “REALIZATION OF THE TRANSFORMATION.” Then,${w \equiv {w\left( {\sigma,{ɛ(\sigma)}} \right)}} = {\left( \frac{L_{1}}{L_{2}} \right)^{3}\frac{{\exp \left( {\quad k\quad L_{2}} \right)}\left( {1 - {\quad k\quad L_{2}}} \right)}{{\exp \left( {\quad k\quad L_{1}} \right)}\left( {1 - {\quad k\quad L_{1}}} \right)}}$

where the wave number k≡k(σ,∈(σ))={square root over (iωμ(σ+iω∈(σ)))},and the dependence of the dielectric constant ∈ on the conductivity σ isaccounted for by the function ∈(σ). Different functions ∈(σ) can beselected for different types of rock. Let σ_(P) and σ_(A) denote twoestimates of the conductivity based on a phase and an attenuationmeasurement and a model such as the above model. The estimates can bedetermined by solving the system of equations

0=|w ₁ |−|w(σ_(A),∈(σ_(P)))|

0=phase(w ₁)−phase(w(σ_(P),∈(σ_(A)))).

The first equation involves the magnitude (a.k.a. the attenuation) ofthe measurement and the second equation involves the phase (a.k.a. thephase shift) of the measurement. Note that the dielectric constant ofone equation is evaluated using the conductivity of the other equation.

This disclosed technique does not make use of the “old assumptions.”Instead, the attenuation conductivity is evaluated using a dielectricvalue consistent with the phase conductivity and the phase conductivityis evaluated using a dielectric constant consistent with the attenuationconductivity. These conductivity estimates are not independent becausethe equations immediately above are coupled (i.e., both variables appearin both equations). The above described techniques represent asubstantial improvement in estimating two resistivity values from aphase and an attenuation measurement given a priori information aboutthe dependence of the dielectric constant on the conductivity. It can beshown that the sensitivity functions for the conductivity estimatesσ_(A)and σ_(P) are S′ and S″, respectively if the perturbation to thevolume P 225 is consistent with the assumed dependence of the dielectricconstant on the conductivity.

It will be evident to those skilled in the art that a more complicatedmodel can be used in place of the simplifying assumptions. Such a modelmay include finite antennas, metal or insulating mandrels, formationinhomogeneities and the like. In addition, other systems of equationscould be defined such as ones involving the real and imaginary parts ofthe measurements and model values. As in previous sections of thisdisclosure, calibration factors and borehole corrections may be appliedto the raw data.

The foregoing disclosure and description of the various embodiments areillustrative and explanatory thereof, and various changes in thedescriptions and attributes of the system, the organization of themeasurements, transmitter and receiver configurations, and the order andtiming of steps taken, as well as in the details of the illustratedsystem may be made without departing from the spirit of the invention.

What we claim is:
 1. A method of determining independent estimates ofelectrical parameters of an earth formation, the method comprising: (a)obtaining a plurality of measured electrical signals that havepenetrated the earth formation, each of said electrical parametersrepresentative of a different property of the earth formation; (b)comparing said measured electrical signals to a model that estimates,independently among at least first and second electrical parameters,said measured electrical signals as a function of electrical parameters;and (c) assigning values to selected electrical parameters such that themodel generates estimated electrical signals that agree with themeasured electrical signals.
 2. The method of claim 1, in which (b) and(c) comprise iterative forward modeling.
 3. The method of claim 2,further comprising: (d) repeating (c) using background values tooptimize successive selections of values for the electrical parameters.4. The method of claim 3, further comprising: (e) solving forperturbation values in the background values to further assist saidoptimization.
 5. The method of claim 4, in which (e) further comprises:computing a plurality of measurement sensitivity values corresponding toa measured electrical signal within the plurality thereof based upon achange, within a preselected volume of the earth formation, in one ormore of the electrical parameters from one or more of the backgroundvalues; using selected measured electrical signals and selectedmeasurement sensitivity values to determine a plurality of perturbationvalues corresponding to variances between the electrical parameters andone or more of the background values; and correlating selectedelectrical parameters to the sum of a corresponding perturbation valueand a corresponding background value.
 6. The method of claim 4, in whichthe perturbation values are determined using a formula:${{\Delta \quad \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\quad \Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right)}}}};$

in which formula: Δ{circumflex over (σ)} is a complex number including areal part and an imaginary part, the real part of Δ{circumflex over (σ)}correlated to the difference between a conductivity value of the earthformation and a conductivity value of a background medium, the imaginarypart of Δ{circumflex over (σ)} correlated to the product of anexcitation frequency and the difference between a dielectric constant ofthe earth formation and a dielectric constant of the background medium;I[•] is an integral operator representing an integral over all space ofan argument for said integral operator I[•]; S is a complex-valuedsensitivity function relating attenuation and phase shift measurementsto variations in electrical parameters as a function of position withinthe earth formation; Δ{tilde over (σ)} is a complex-valued function ofposition within the earth formation including real and imaginary parts,the real part of Δ{tilde over (σ)} representing the difference between aconductivity value of the earth formation and a conductivity value ofthe background medium at a predetermined location in a volume ofinterest, the imaginary part of Δ{tilde over (σ)} proportional to thedifference between a dielectric constant of the earth formation and adielectric constant of the background medium at the predeterminedlocation; c is a calibration factor comprising a complex numberaccounting for irregularities in excitation of the measured electricalsignals and in measurement thereof; w₁ is a complex number representingthe plurality of measured electrical signals; w₀ represents an expectedvalue of w₁ in the background medium; and w_(bh) is a complex numberaccounting for borehole effects when the plurality of measuredelectrical signals is obtained from a borehole.
 7. The method of claim1, in which the at least first and second electrical parameters includeat least one electrical parameter selected from the group consisting ofa dielectric constant and a resistivity value.
 8. The method of claim 1,in which the measured electrical signals have a frequency between 5 kHzand 2 GHz.
 9. The method of claim 1, in which the measured electricalsignals include at least one measurement selected from the groupconsisting of an attenuation measurement and a phase shift measurement.10. The method of claim 1, in which measured electrical signals arederived from a predetermined ratio of voltages measured by at least tworeceiver antennas.
 11. The method of claim 1, in which the measuredelectrical signals originate from selected points along a borehole. 12.The method of claim 1, further comprising: (d) repeating (c) usingbackground values to optimize successive selections of values for theelectrical parameters; and (e) compensating selected background valuesfor borehole errors.
 13. The method of claim 1, further comprising: (d)compensating selected measured electrical signals for borehole errors.14. A method of determining independent estimates of electricalparameters of an earth formation, the method comprising: (a) obtaining aplurality of measured electrical signals that have penetrated the earthformation, each of said electrical parameters representative of adifferent property of the earth formation; (b) comparing said measuredelectrical signals to a model that estimates, independently among atleast first and second electrical parameters, said measured electricalsignals as a function of electrical parameters; (c) assigning values toselected electrical parameters such that the model generates estimatedelectrical signals that agree with the measured electrical signal; (d)repeating (c) using background values to optimize successive selectionsof values for the electrical parameters; and (e) solving forperturbation values in the background values.
 15. The method of claim14, in which (b) and (c) comprise iterative forward modeling.
 16. Themethod of claim 14, in which (e) further comprises: computing aplurality of measurement sensitivity values corresponding to a measuredelectrical signal within the plurality thereof based upon a change,within a preselected volume of the earth formation, in one or more ofthe electrical parameters from one or more of the background values;using selected measured electrical signals and selected measurementsensitivity values to determine a plurality of perturbation valuescorresponding to variances between the electrical parameters and one ormore of the background values; and correlating selected electricalparameters to the sum of a corresponding perturbation value and acorresponding background value.
 17. The method of claim 14, in which theperturbation values are determined using a formula:${{\Delta \quad \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\quad \Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right)}}}};$

in which formula: Δ{circumflex over (σ)} is a complex number including areal part and an imaginary part, the real part of Δ{circumflex over (σ)}correlated to the difference between a conductivity value of the earthformation and a conductivity value of a background medium, the imaginarypart of Δ{circumflex over (σ)} correlated to the product of anexcitation frequency and the difference between a dielectric constant ofthe earth formation and a dielectric constant of the background medium;I[•] is an integral operator representing an integral over all space ofan argument for said integral operator I[•]; S is a complex-valuedsensitivity function relating attenuation and phase shift measurementsto variations in electrical parameters as a function of position withinthe earth formation; Δ{tilde over (σ)} is a complex-valued function ofposition within the earth formation including real and imaginary parts,the real part of Δ{tilde over (σ)} representing the difference between aconductivity value of the earth formation and a conductivity value ofthe background medium at a predetermined location in a volume ofinterest, the imaginary part of Δ{tilde over (σ)} proportional to thedifference between a dielectric constant of the earth formation and adielectric constant of the background medium at the predeterminedlocation; c is a calibration factor comprising a complex numberaccounting for irregularities in excitation of the measured electricalsignals and in measurement thereof; w₁ is a complex number representingthe plurality of measured electrical signals; w₀ represents an expectedvalue of w₁ in the background medium; and w_(bh) is a complex numberaccounting for borehole effects when the plurality of measuredelectrical signals is obtained from a borehole.
 18. The method of claim14, in which the at least first and second electrical parameters includeat least one electrical parameter selected from the group consisting ofa dielectric constant and a resistivity value.
 19. The method of claim14, which the measured electrical signals include at least onemeasurement selected from the group consisting of an attenuationmeasurement and a phase shift measurement.
 20. The method of claim 14,in which measured electrical signals are derived from a predeterminedratio of voltages measured by at least two receiver antennas.
 21. Themethod of claim 14, further comprising: (f) compensating selectedbackground values for borehole errors.
 22. The method of claim 14,further comprising: (f) compensating selected measured electricalsignals for borehole errors.
 23. A computer system comprising: at leastone processor programmed to execute a method of determining independentestimates of electrical parameters of an earth formation, the methodcomprising: (a) obtaining a plurality of measured electrical signalsthat have penetrated the earth formation, each of said electricalparameters representative of a different property of the earthformation; (b) comparing said measured electrical signals to a modelthat estimates, independently among at least first and second electricalparameters, said measured electrical signals as a function of electricalparameters; and (c) assigning values to selected electrical parameterssuch that the model generates estimated electrical signals that agreewith the measured electrical signals.
 24. The computer system of claim23, in which (b) and (c) in the method comprise iterative forwardmodeling.
 25. The computer system of claim 23, in which the methodfurther comprises: (d) repeating (c) using background values to optimizesuccessive selections of values for the electrical parameters.
 26. Thecomputer system of claim 25, in which the method further comprises: (e)solving for perturbation values in the background values to furtherassist said optimization.
 27. The computer system of claim 26, in which(e) in the method further comprises: computing a plurality ofmeasurement sensitivity values corresponding to a measured electricalsignal within the plurality thereof based upon a change, within apreselected volume of the earth formation, in one or more of theelectrical parameters from one or more of the background values; usingselected measured electrical signals and selected measurementsensitivity values to determine a plurality of perturbation valuescorresponding to variances between the electrical parameters and one ormore of the background values; and correlating selected electricalparameters to the sum of a corresponding perturbation value and acorresponding background value.
 28. The computer system of claim 26, inwhich the method determines perturbation values using a formula:${{\Delta \quad \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\quad \Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right)}}}};$

in which formula: Δ{circumflex over (σ)} is a complex number including areal part and an imaginary part, the real part of Δ{circumflex over (σ)}correlated to the difference between a conductivity value of the earthformation and a conductivity value of a background medium, the imaginarypart of Δ{circumflex over (σ)} correlated to the product of anexcitation frequency and the difference between a dielectric constant ofthe earth formation and a dielectric constant of the background medium;I[•] is an integral operator representing an integral over all space ofan argument for said integral operator I[•]; S is a complex-valuedsensitivity function relating attenuation and phase shift measurementsto variations in electrical parameters as a function of position withinthe earth formation; Δ{tilde over (σ)} is a complex-valued function ofposition within the earth formation including real and imaginary parts,the real part of Δ{tilde over (σ)} representing the difference between aconductivity value of the earth formation and a conductivity value ofthe background medium at a predetermined location in a volume ofinterest, the imaginary part of Δ{tilde over (σ)} proportional to thedifference between a dielectric constant of the earth formation and adielectric constant of the background medium at the predeterminedlocation; c is a calibration factor comprising a complex numberaccounting for irregularities in excitation of the measured electricalsignals and in measurement thereof; w₁ is a complex number representingthe plurality of measured electrical signals; w₀ represents an expectedvalue of w₁ in the background medium; and w_(bh) is a complex numberaccounting for borehole effects when the plurality of measuredelectrical signals is obtained from a borehole.
 29. The computer systemof claim 23, in which the at least first and second electricalparameters include at least one electrical parameter selected from thegroup consisting of a dielectric constant and a resistivity value. 30.The computer system of claim 23, in which the measured electricalsignals include at least one measurement selected from the groupconsisting of an attenuation measurement and a phase shift measurement.31. The computer system of claim 23, in which measured electricalsignals are derived from a predetermined ratio of voltages measured byat least two receiver antennas.
 32. The computer system of claim 23, inwhich the method further comprises: (d) repeating (c) using backgroundvalues to optimize successive selections of values for the electricalparameters; and (e) compensating selected background values for boreholeerrors.
 33. The computer system of claim 23, in which the method furthercomprises: (d) compensating selected measured electrical signals forborehole errors.
 34. A computer system comprising: at least oneprocessor; and a storage device having computer-readable logic storedtherein, the computer-readable logic accessible by and intelligible tothe processor; the computer-readable logic further configured toinstruct the processor to execute a method of determining independentestimates of electrical parameters of an earth formation, the methodcomprising: (a) obtaining a plurality of measured electrical signalsthat have penetrated the earth formation, each of said electricalparameters representative of a different property of the earthformation; (b) comparing said measured electrical signals to a modelthat estimates, independently among at least first and second electricalparameters, said measured electrical signals as a function of electricalparameters; (c) assigning values to selected electrical parameters suchthat the model generates estimated electrical signals that agree withthe measured electrical signals; (d) repeating (c) using backgroundvalues to optimize successive selections of valves for the electricalparameters; and (e) solving for perturbation values in the backgroundvalues.
 35. The computer system of claim 34, in which (b) and (c) in themethod comprise iterative forward modeling.
 36. The computer system ofclaim 34, in which (e) in the method further comprises: computing aplurality of measurement sensitivity values corresponding to a measuredelectrical signal within the plurality thereof based upon a change,within a preselected volume of the earth formation, in one or more ofthe electrical parameters from one or more of the background values;using selected measured electrical signals and selected measurementsensitivity values to determine a plurality of perturbation valuescorresponding to variances between the electrical parameters and one ormore of the background values; and correlating selected electricalparameters to the sum of a corresponding perturbation value and acorresponding background value.
 37. The computer system of claim 34, inwhich the method determines perturbation values using a formula:${{\Delta \quad \hat{\sigma}} = {\frac{I\left\lbrack {S\quad \Delta \overset{\sim}{\sigma}} \right\rbrack}{I\lbrack S\rbrack} = {{I\left\lbrack {\hat{S}\quad \Delta \overset{\sim}{\sigma}} \right\rbrack} = {\frac{1}{I\lbrack S\rbrack}\left( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} \right)}}}};$

in which formula: Δ{circumflex over (σ)} is a complex number including areal part and an imaginary part, the real part of Δ{circumflex over (σ)}correlated to the difference between a conductivity value of the earthformation and a conductivity value of a background medium, the imaginarypart of Δ{circumflex over (σ)} correlated to the product of anexcitation frequency and the difference between a dielectric constant ofthe earth formation and a dielectric constant of the background medium;I[•] is an integral operator representing an integral over all space ofan argument for said integral operator I[•]; S is a complex-valuedsensitivity function relating attenuation and phase shift measurementsto variations in electrical parameters as a function of position withinthe earth formation; Δ{tilde over (σ)} is a complex-valued function ofposition within the earth formation including real and imaginary parts,the real part of Δ{tilde over (σ)} representing the difference between aconductivity value of the earth formation and a conductivity value ofthe background medium at a predetermined location in a volume ofinterest, the imaginary part of Δ{tilde over (σ)} proportional to thedifference between a dielectric constant of the earth formation and adielectric constant of the background medium at the predeterminedlocation; c is a calibration factor comprising a complex numberaccounting for irregularities in excitation of the measured electricalsignals and in measurement thereof; w₁ is a complex number representingthe plurality of measured electrical signals; w₀ represents an expectedvalue of w₁ in the background medium; and w_(bh) is a complex numberaccounting for borehole effects when the plurality of measuredelectrical signals is obtained from a borehole.
 38. The computer systemof claim 34, in which the at least first and second electricalparameters include at least one electrical parameter selected from thegroup consisting of a dielectric constant and a resistivity value. 39.The computer system of claim 34, in which the measured electricalsignals include at least one measurement selected from the groupconsisting of an attenuation measurement and a phase shift measurement.40. The computer system of claim 34, in which measured electricalsignals are derived from a predetermined ratio of voltages measured byat least two receiver antennas.
 41. The computer system of claim 34, inwhich the method further comprises: (f) compensating selected backgroundvalues for borehole errors.
 42. The computer system of claim 34, inwhich the method further comprises: (f) compensating selected measuredelectrical signals for borehole errors.
 43. A computer systemcomprising: at least one processor; and a storage device havingcomputer-readable logic stored therein, the computer-readable logicaccessible by and intelligible to the processor; the computer-readablelogic further configured to instruct the processor to execute a methodof determining independent estimates of electrical parameters of anearth formation, the method comprising: (a) obtaining a plurality ofmeasured electrical signals that have penetrated the earth formation,each of said electrical parameters representative of a differentproperty of the earth formation; (b) comparing said measured electricalsignals to a model that estimates, independently among at least firstand second electrical parameters, said measured electrical signals as afunction of electrical parameters; and (c) assigning values to selectedvalues electrical parameters such that the model generates estimatedelectrical signals that agree with the measured electrical signals. 44.A computer system comprising: at least one processor programmed toexecute a method of determining independent estimates of electricalparameters of an earth formation, the method comprising: (a) obtaining aplurality of measured electrical signals that have penetrated the earthformation, each of said electrical parameters representative of adifferent property of the earth formation; (b) comparing said measuredelectrical signals to a model that estimates, independently among atleast first and second electrical parameters, said measured electricalsignals as a function of electrical parameters; (c) assigning values toselected electrical parameters such that the model generates estimatedelectrical signals that agree with the measured electrical signals; (d)repeating (c) using background values to optimize successive selectionsof values for the electrical parameters; and (e) solving forperturbation values in the background values.